Download Exact Values of Trigonometric Ratios

trigonometry
trigonometry
Special Angles
MCR3U: Functions
Consider a right-isosceles triangle where the two equal sides
have a length of 1 unit.
Exact Values of Trigonometric Ratios
J. Garvin
Since an isosceles triangle has two equal angles, the acute
angles must both be 45◦ .
J. Garvin — Exact Values of Trigonometric Ratios
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trigonometry
Special Angles
trigonometry
Special Angles
Using√the Pythagorean
Theorem for the hypotenuse,
√
h = 12 + 12 = 2.
sin 45◦ =
=
csc 45◦ =
=
√1
2
√
2
2
cos 45◦ =
=
tan 45◦ =
1
1
=1
and
√
2
1
sec 45◦ =
2
=
√
√1
2
√
2
2
√
2
√1
cot 45◦ =
2
1
1
=1
The resulting triangle can be used to determine exact values
for trigonometric ratios involving 45◦ .
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trigonometry
trigonometry
Special Angles
Special Angles
Now consider an equilateral triangle with side lengths of 2
units.
An altitude from one vertex creates two congruent
right-triangles with 30◦ and 60◦ angles, one side 1 unit long,
and a hypotenuse 2 units long.
Using√the Pythagorean
Theorem for the other side,
√
a2 = 22 − 12 = 3.
An equilateral triangle contains three 60◦ angles.
This triangle can be used to find exact values for
trigonometric ratios involving 30◦ or 60◦ .
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trigonometry
Special Angles
sin 30◦ =
1
2
Angles in Quadrant 1
cos 30◦ =
√
3
2
tan 30◦ =
√
2 3
3
cot 30◦ =
and
csc 30◦ = 2
sin 60◦ =
csc 60◦ =
√
3
2
√
2 3
3
trigonometry
sec 30◦ =
cos 60◦ =
1
2
tan 60◦ =
√
3
3
√
√
and
sec 60◦ = 2
cot 60◦ =
3
Recall that two angles are coterminal if their terminal arms
are in the same position on the coordinate plane.
For example, 405◦ is coterminal with 45◦ , since
405◦ − 360◦ = 45◦ .
This means that the ratios for 405◦ are the same as those for
45◦ .
For instance, sin 405◦ = sin 45◦ =
3
√
2
2 .
Negative rotations may also result in coterminal angles in Q1.
√
3
3
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trigonometry
trigonometry
Angles in Quadrant 1
Angles in Quadrants 2-4
Example
When an angle lies in quadrants 2-4, we use the reference
angle – the acute angle made with the x-axis.
Determine the exact value of sin −330◦ .
By the CAST rule, trigonometric ratios may be positive or
negative, depending on the quadrant in which the angle falls.
A rotation of −330◦ is the same as a rotation of 30◦ .
When used in conjunction with the CAST rule, it is possible
to determine the exact values of trigonometric ratios in the
other three quadrants, including their sign.
Using the 30-60-90 triangle, the opposite side has a length of
1 and the hypotenuse has a length of 2.
opp
Therefore, sin 30◦ =
= 12 .
hyp
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trigonometry
trigonometry
Angles in Quadrants 2-4
Angles in Quadrants 2-4
Example
Since sine is negative in Q3, the exact value will be too.
Determine the exact value of sin 210◦ .
210◦
falls in quadrant 3, with a reference angle of
Therefore, sin 210◦ = − sin 30◦ .
30◦ .
Using the 30-60-90 triangle, sin 30◦ = 12 .
So, sin 210◦ = − 12 .
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trigonometry
trigonometry
Angles in Quadrants 2-4
Angles in Quadrants 2-4
Example
Since cosine is positive in Q4, the exact value will be too.
Determine the exact value of cos 315◦ .
Therefore, cos 315◦ = cos 45◦ .
315◦ falls in quadrant 4, with a reference angle of 45◦ .
Using the 45-45-90 triangle, cos 45◦ =
So,
cos 315◦
=
√
√
2
2 .
2
2 .
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trigonometry
trigonometry
Angles in Quadrants 2-4
Angles in Quadrants 2-4
Example
Since cosecant is negative in Q3, the exact value will be too.
Determine the exact value of csc(−120◦ ).
Therefore, csc(−120◦ ) = − csc 60◦ .
−120◦ falls in quadrant 3, with a reference angle of 60◦ .
Using the 30-60-90 triangle, csc 60◦ =
√
√
2 3
3 .
So, csc(−120◦ ) = − 2 3 3 .
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trigonometry
trigonometry
Angles Falling Between Quadrants
Angles Falling Between Quadrants
Sometimes, the terminal arm of an angle falls on an axis,
between two quadrants.
Example
The following table summarizes the exact values of the
primary trigonometric ratios for 0◦ , 90◦ , 180◦ and 270◦ .
Since 630◦ is coterminal with 270◦ as shown,
cos 630◦ = cos 270◦ = 0.
Angle
0◦
90◦
180◦
270◦
sin θ
cos θ
tan θ
0
1
0
1
0
undef.
0
−1
0
−1
0
undef.
Determine the exact value of cos 630◦ .
These values will take on more meaning in the next unit on
trigonometric functions.
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trigonometry
Questions?
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